Question

If the sum of first 8 terms of AP is 136 and that of first 15 terms is 465 then find the sum of first 25 terms.​

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{Sum\:of\:first\:25th\:term=1275}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline \bold{Given :}} \\ \tt: \implies Sum \: of \: first \: 8 \: terms(s_{8}) = 136 \\ \\ \tt: \implies Sum \: of \: first \: 15 \: terms(s_{15}) = 465 \\ \\\red{\underline \bold{To \: Find:}} \\ \tt: \implies Sum \: of \: first \: 25 \: terms(s_{25}) = ?$

• According to given question :

$\bold{As \: we \: know \: that} \\ \tt: \implies s_{n} = \frac{n}{2} (2a +(n - 1)d) \\ \\ \tt: \implies s_{8} = \frac{8}{2} (2a + (8 - 1) \times d) \\ \\ \tt: \implies 136 = 4 \times (2a + 7d) \\ \\ \tt: \implies 2a + 7d = 34 - - - - - (1) \\ \\ \bold{Similarly} \\ \tt: \implies s_{15} = \frac{15}{2} (2a + (15 - 1) \times d) \\ \\ \tt: \implies 465 \times \frac{2}{15} = 2a + 14d \\ \\ \tt: \implies 2a + 14d = 62 - - - - - (2) \\ \\ \text{Subtracting \: (1) \: from \: (2)} \\ \tt: \implies 14d - 7d = 62 - 34 \\ \\ \tt: \implies 7d = 28 \\ \\ \green{\tt: \implies d = 4}$

$\text{Putting \: value \: of \: d \: in \: (1)} \\ \tt: \implies 2a + 7d = 34 \\ \\ \tt: \implies 2a = 34 - 28\\ \\ \green{\tt: \implies a = 3} \\ \\ \bold{For \:sum \: of \: first \: 25th \: term( \: s_{25})} \\ \tt: \implies s_{25} = \frac{25}{2} (2 \times 3 + (25 - 1) \times 4) \\ \\ \tt: \implies s_{25} = \frac{25}{2} (6 + 96) \\ \\ \tt: \implies s_{25} = \frac{25}{2} \times 102 \\ \\ \green{\tt: \implies s_{25} =1275} \\ \\ \green{\tt{\therefore Sum \: of \: first \: 25th \: term \: is \: \: 1275}}$

Answer 2

AnswEr:

$\small\bold{\underline{\sf{\blue{\:\:Given\;\::-}}}}$

Sum of the 8th Terms is 136 and that of first 15 terms is 465.

To Find:-

Sum of first 25 Terms = ?

Explanation:-

We know that,

$\bigstar\;\:\small{\underline{\boxed{\sf{\red{\dfrac{n}{2} 2a \:+\:(n\:-\:1)d}}}}}$

Here,

• a = First Term
• D = Common Difference

$\bold{\underline{\sf{According\:to\:Ques\: Now}}}$

$\longrightarrow\sf\: \dfrac{8}{2} (2a \:+\:(8\:-\:1)d \:=\: 136$

$\longrightarrow\sf\: 2a + 7d = 136$ -------Eq(1)

$\rule{150}2$

$\longrightarrow\sf\: \dfrac{15}{2} (2a \:+\:(15\:-\:1)d = 465$

$\longrightarrow\sf\: 2a \:+\:14d \:=\:\dfrac{465\times\: 2}{15}$

$\longrightarrow\sf\: 2a + 14d = 62$ -------Eq(2)

$\rule{150}2$

$\dag\:\small\bold{\underline{\sf{\blue{Now,\: Subtracting\: Equation\:(1)\:from\:(2)}}}}$

$\longrightarrow\sf\: 2a \: + \: 17d \: =\: 136$

$\longrightarrow\sf\:2a \:+\: 14d \:=\: 62$

$\longrightarrow\sf\: - 7d = - 28$

$\longrightarrow\sf\: d = \dfrac{-28}{-7}$

$\longrightarrow\large\boxed{\sf{\red{d\:=\: 4}}}$

★ Substituting the Value of d in Equation (1)

$\longrightarrow\sf\: 2a + 7d = 34$

$\longrightarrow\sf\:2a + 7(4) = 34$

$\longrightarrow\sf\:2a + 28 = 34$

$\longrightarrow\sf\:2a = 34 - 28$

$\longrightarrow\sf\:2a = 6$

$\longrightarrow\sf\:a = \cancel\dfrac{6}{2}$

$\longrightarrow\large{\sf{\red{a \:=\: 3}}}$

$\rule{150}2$

Now, Finding the Sum of 25 Terms

$\longrightarrow\sf\: Sn = \dfrac{n}{2}(2a \:+\;(n \:-\:1)d$

$\longrightarrow\sf\: = \dfrac{25}{2}(6 + 24 \times \: 4)$

$\longrightarrow\sf\: \dfrac{25}{2} \times\: 102$

$\longrightarrow\large{\underline{\boxed{\sf{\pink{1275}}}}}$

$\small\bold{\underline{\sf{\blue{Hence,\;Sum\:of\: First\:25\;terms\: is\:1275}}}}$